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Mirrors > Home > MPE Home > Th. List > Mathboxes > ttrclco | Structured version Visualization version GIF version |
Description: Composition law for the transitive closure of a relationship. (Contributed by Scott Fenton, 20-Oct-2024.) |
Ref | Expression |
---|---|
ttrclco | ⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5877 | . . . 4 ⊢ Rel (𝑅 ↾ V) | |
2 | ssttrcl 33511 | . . . 4 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)) | |
3 | coss2 5722 | . . . 4 ⊢ ((𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) → (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V))) | |
4 | 1, 2, 3 | mp2b 10 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) |
5 | ttrcltr 33512 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) | |
6 | 4, 5 | sstri 3907 | . 2 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) |
7 | relco 6105 | . . . 4 ⊢ Rel (t++(𝑅 ↾ V) ∘ 𝑅) | |
8 | dfrel3 6058 | . . . 4 ⊢ (Rel (t++(𝑅 ↾ V) ∘ 𝑅) ↔ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ 𝑅)) | |
9 | 7, 8 | mpbi 233 | . . 3 ⊢ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ 𝑅) |
10 | resco 6111 | . . 3 ⊢ ((t++(𝑅 ↾ V) ∘ 𝑅) ↾ V) = (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) | |
11 | ttrclresv 33513 | . . . 4 ⊢ t++(𝑅 ↾ V) = t++𝑅 | |
12 | 11 | coeq1i 5725 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ 𝑅) = (t++𝑅 ∘ 𝑅) |
13 | 9, 10, 12 | 3eqtr3i 2773 | . 2 ⊢ (t++(𝑅 ↾ V) ∘ (𝑅 ↾ V)) = (t++𝑅 ∘ 𝑅) |
14 | 6, 13, 11 | 3sstr3i 3940 | 1 ⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 Vcvv 3405 ⊆ wss 3863 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 t++cttrcl 33503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rmo 3066 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-int 4857 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-oadd 8203 df-ttrcl 33504 |
This theorem is referenced by: (None) |
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